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A Yosida–Hewitt decomposition for minitive set functions
Fuzzy Sets and Systems 157 (2006) 308 – 318 www.elsevier.com/locate/fss
A Yosida–Hewitt decomposition for minitive set functions Yann Rébillé∗ Université Paris I, CERMSEM, 106-112 boulevard de l’Hopital, 75647 Paris Cedex 13, France Received 25 August 2004; received in revised form 9 June 2005; accepted 17 June 2005 Available online 11 July 2005
Abstract We prove for minitive set functions defined on a -algebra, a similar decomposition theorem to the Yosida–Hewitt’s one for classical measures, this way any minitive set function can be decomposed in a fuzzy minitive measure part and a purely minitive part. A particular attention is given for the countable case where the canonical description of any -continuous necessity is fully elicited and provide a simple way to compute the Choquet integral of any bounded sequence. © 2005 Elsevier B.V. All rights reserved. Keywords: Non-additive measures; Yosida–Hewitt decomposition; Necessity measures; Choquet integral
1. Introduction A possibility measure [13] is a set function : P () −→ [0, 1] such that (∅) = 0, () = 1 and for all family of subsets {Ai }I it holds, (∪I Ai ) = supI (Ai ), where the index set I is arbitrary. This last property expresses the complete maxitivity of the set function . By duality one can define a necessity measure associated to through N(A) = 1 − (Ac ) (where (.)c denotes the complement). In this case the complete maxitivity translates into complete minitivity i.e. for all family of subsets {Ai }I it holds, N(∩I Ai ) = inf I N(Ai ). A fuzzy measure [11] v : A −→ [0, 1] is a set function defined on a -algebra which satisfies the following conditions v(∅) = 0 , A ⊂ B ⇒ v(A) v(B), A1 ⊂ A2 ⊂ . . . , ⇒ v(∪n An ) = lim∞ v(An ) and A1 ⊃ A2 ⊃ . . . , ⇒ v(∩n An ) = lim∞ v(An ). Puri and Ralescu [9] proved that not all possibilities are fuzzy measures and that possibilities which are fuzzy measures are ∗ Tel.: +33 625454442.
E-mail address: [email protected]. 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.06.012
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ill-behaved on metric spaces. Nguyen [8] suggested for the case of topological space to restrict continuity from above to countable sequences of compact set, this way a Choquet [4] capacity would obtain if we start from an upper semi-continuous density. A natural restriction to the possibility measure is to maintain the maxitivity property only for finite index sets. This way the density function defined for all ∈ through () = ({}) does not summarize all the information. Now in order to obtain continuity from below it is sufficient and necessary to extend the maxitivity property to countable index set, but this last condition does not guarantee continuity from above, thus might not be a fuzzy measure. In the context of classical measure theory [12] provide an important theorem that allows one to decompose any finitely additive set function P into a -additive part and a pathological part termed purely finitely additive i.e. for all -additive set function, 0 P ⇒ = 0. Theorem (Yosida and Hewitt [12]). Let be a finitely additive set function on (, A) then there exists a unique pair of additive set functions (c , p ) with c -additive and p purely finitely additive such that = c + p . Our aim is to give a similar decomposition for the case of necessity measures and decompose any necessity set function into a fuzzy measure part and a purely minitive part. The paper is structured as follows. In Section 2 we give the definition and notations relative to set functions and list some properties of minitive set function that will be used in the sequel. In Section 3 the general case is treated through the use of the canonical description of minitive set functions. In Section 4 we focus on the countable case where the canonical description of any -continuous necessity is fully elicited and provide a simple way to compute the Choquet integral of any bounded sequence.
2. Definitions, notations Let be a non-empty set, and A a -algebra of subsets of . A set function v on A will always be real valued and zero at the empty set i.e. v : A −→ R, v(∅) = 0. Such set functions are known in the field of cooperative games as games. A set function is said to be monotone if ∀A, B ∈ A, A ⊂ B ⇒ v(A) v(B); normalized if v() = 1. A set function v is said to be convex if v is a monotone set function and supermodular i.e. ∀A, B ∈ A, v(A∪B)+v(A∩B) v(A)+v(B). A non-negative set function v (i.e. v 0) is said to be a (finitely) additive set function if ∀A, B ∈ A, A ∩ B = ∅, v(A ∪ B) = v(A) + v(B). If furthermore v() = 1, v is a probability. A set function is said to be -additive if ∀ {An }n ⊂ A, with An ∩ Am = ∅, when m = n, v(∪n An ) = n v(An ). In particular an additive set function is convex. A set function is said to be minitive if ∀A, B ∈ A, v(A ∩ B) = min{v(A), v(B)}. If v is normalized then v is called a necessity (see [6]). It turns out immediately that v is convex. A set function is said to be continuous from below if for all non-decreasing sequences {An }n ⊂ A, lim∞ v(An ) = v(∪n An ). A set function is said to be continuous from above if for all non-increasing sequences {Bn }n ⊂ A, lim∞ v(Bn ) = v(∩n Bn ). v is said to be -continuous if for all sequence {An }n ⊂ A converging to A ∈ A i.e. such that A = ∩n ∪k n Ak = ∪n ∩k n Ak we have lim∞ v(An ) = v(A). If v is a monotone set function then -continuity is equivalent to continuity from below and above. Monotone -continuous set functions are known as fuzzy measures (see [9]). For the case of additive measures these definitions of continuity collapse to -additivity.
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A simple way to obtain -continuity of a set function is through convexity, if v is convex then v is -continuous as soon as v is continuous at i.e. for all non-decreasing sequences {An }n ⊂ A such that ∪n An = , lim∞ v(An ) = v() (see Appendix). The definition of purity for additive set functions can be adapted in the same way to minitive set functions, Definition 2.1. A minitive set function v is said to be pure if for all non-negative -continuous set function, 0 v ⇒ = 0. Let v be a monotone set function on A, define for any X ∈ B∞ (, A) the Choquet integral [4] of X with respect to v as, ∞ 0 X dv = v({X t}) d(t) + (v({X t}) − v()) dt, 0
−∞
where {X t} = { ∈ : X() t} and B∞ (, A) denotes the set of bounded (w.r.t. the supremum norm) A-measurable functions. A family F of subsets of with F ⊂ A is said to be a filter (e.g. [1]) if, (i) ∅ ∈ / F, ∈ F, (ii) ∀A, B ∈ A, [A, B ∈ F ⇒ A ∩ B ∈ F ], (iii) ∀A, B ∈ A, [A ∈ F , A ⊂ B ⇒ B ∈ F ]. A filter is termed -filter if it satisfies the following condition: (iv) for all non-decreasing sequence {An }n with ∪n An = , there exists N ∈ N such that AN ∈ F . Another way to state (iv) is the following, (iv ) Let {An }n ⊂ A be a countable partition i.e. ∀n = m, An ∩ Am = ∅ and ∪n An = then there exists a finite set I ⊂ N such that ∪n∈I An ∈ F . This way if F is a -filter and {An }n ⊂ A a countable partition then one can find a (minimal) finite set I0 such that ∪n∈I0 An ∈ F and ∀I ⊂ N, ∪n∈I An ∈ F ⇒ I0 ⊂ I . Let F be a filter on , define the filter game uF , 1 if B ∈ F , uF (B) = 0 otherwise. For A ∈ A, A = ∅ let FA = {B : B ∈ A, A ⊂ B} we define the unanimity game uA on A by uA := uFA . Notice that for a filter game uF , uF is continuous from below (hence -continuous) if and only if F is a -filter. The nature of a filter game is clear-cut with respect to continuity, Lemma 1. Let uF be a filter game, then uF is pure if and only if uF is not -continuous. Proof. (only if) Let uF be pure and assume uF -continuous, we have uF uF hence uF = 0, what is impossible. (if) Let be -continuous with 0 uF . Since uF is not -continuous, there exists a non-decreasing sequence {An }n with ∪n An = such that ∀n ∈ N, uF (An ) = 0. Now let A ∈ A, for all n ∈ N we have 0 (A ∩ An ) uF (A ∩ An ) uF (An ) = 0 and since {A ∩ An }n converges to A -continuity of entails (A) = 0, so uF is pure.
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Filter games play a keyrole in the integral representation of belief functions (see [7]) since they constitute the set of extreme points of the (convex) set of belief functions (see [2,3]). It appears that they will also play a keyrole in the representation of necessities, since any filter game is a necessity, despite the fact that the set of necessities is not convex. We state some properties that will be useful in the sequel, Properties. The following statements hold, (i) Let v be a minitive set function and > 0 then . v is a minitive set function. (ii) Let v1 , v2 be minitive set functions then v1 + v2 is a minitive set function if and only if v1 , v2 agree i.e. ∃A, B ∈ A s.t. v1 (A) − v1 (B) > 0 and v2 (A) − v2 (B) < 0. (iii) Let I be an interval of [0, 1), and let {v }∈I be minitive set functions such that v v and v , v agree whenever < and supI v () < +∞ then I v d is a minitive set function. (iv) Let v be a minitive set function and v1 , v2 monotone set functions such that v = v1 + v2 then v1 , v2 are minitive set functions. (v) Let {vi }ni=1 be minitive set functions on (, A) where n ∈ N, = 0 that agree i.e. ∀i, j ∈ {1, . . . , n}, vi , vj agree, and a function : (R+ )n −→ R+ with (0, . . . , 0) = 0 and non-decreasing w.r.t. each of his argument then (v1 , . . . , vn ) is a minitive set function. Moreover {vi }ni=1 ∪ {(v1 , . . . , vn )} agree. Proof of Properties. (i) is immediate; (ii) Let v1 , v2 be minitive set functions that agree. Let A, B be in A, we have (v1 (A) − v1 (B))(v2 (A) − v2 (B)) 0 since v1 , v2 agree and minitivity of v1 , v2 gives v1 (A ∩ B) + v2 (A ∩ B) = Min{v1 (A), v1 (B)} + Min{v2 (A), v2 (B)} = Min{v1 (A) + v2 (A), v1 (B) + v2 (B)}. For the converse let A, B be in A. We start from: Min{v1 (A), v1 (B)} + Min{v2 (A), v2 (B)} = Min{v1 (A) + v2 (A), v1 (B) + v2 (B)}. Suppose first that v1 (A) + v2 (A) < v1 (B) + v2 (B). Now if v1 (A) < v1 (B) then v1 (A)+Min{v2 (A), v2 (B)} = v1 (A)+v2 (A) thus v2 (A) v2 (B); if v2 (A) < v2 (B) then Min{v1 (A), v1 (B)} + v2 (A) = v1 (A) + v2 (A) thus v1 (A) v1 (B). Now if v1 (A) + v2 (A) > v1 (B) + v2 (B) we get same conclusion by symmetry. Finally suppose v1 (A) + v2 (A) = v1 (B) + v2 (B), then Min{v1 (A), v1 (B)} + Min{v2 (A), v2 (B)} = v1 (A) + v2 (A) = v1 (B) + v2 (B) thus v1 (A) = v1 (B) and v2 (A) = v2 (B). So in any case we have shown that (v1 (A) − v1 (B))(v2 (A) − v2 (B)) 0. (iii) Let I be an interval of (0, 1), and let {v }∈I be minitive set functions such that v v and v , v agree whenever < and supI v () < +∞. Without loss of generality we fix I = (0, 1) and supI v () = 1. Set v = I v d. Define v0 (A) = 1 and v1 (A) = 0 for all A ∈ A. Now for all A ∈ A we have that ( → v (A)) is a non-increasing function bounded by 0 and 1, thus we can deal with its Riemann integral on [0, 1]. Define for all n ∈ N, = 0 the set function wn by wn = 1/n ni=1 vi/n . These set functions are minitive from (i), (ii) and converge setwise to v. Let be A, B ∈ A. We have, v(A ∩ B) = lim wn (A ∩ B) n
= lim Min{wn (A), wn (B)} n
= Min{lim wn (A), lim wn (B)} n
n
= Min{v(A), V (B)}.
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Now since this equality is obtained for all A, B ∈ A, v is minitive. (iv) Let v be a minitive set function on (, A) and (v1 , v2 ) a decomposition where v1 , v2 are monotone set functions. Let A, B ∈ A, we have v(A ∩ B) = Min{v(A), v(B)} = Min{v1 (A) + v2 (A), v1 (B) + v2 (B)} Min{v1 (A)), v1 (B)} + Min{v2 (A), v2 (B)} v1 (A ∩ B) + v2 (A ∩ B) = v(A ∩ B) thus Min{v1 (A)), v1 (B)} = v1 (A ∩ B) and Min{v2 (A)), v2 (B)} = v2 (A ∩ B) hold. (v) Let {vi }ni=1 be minitive set functions on (, A) where n ∈ N, = 0 that agree, and a function : (R+ )n −→ R+ with (0, . . . , 0) = 0 and non-decreasing w.r.t. each of his argument. Let us show that (v1 , . . . , vn ) is a minitive set function. Let A, B be in A. If for all i ∈ {1, . . . , n} vi (A) = vi (B) holds we are done since vi (A ∩ B) = vi (A) = vi (B). Now assume without loss of generality that v1 (A) < v1 (B). Since {vi }ni=1 agree then for i = 2, . . . , n we have vi (A) vi (B) so by non-decreasingness of it gives (v1 (A), . . . , vn (A)) (v1 (B), . . . , vn (B)) thus (v1 (A∩B), . . . , vn (A∩B)) = (v1 (A), . . . , vn (A)) so (v1 , . . . , vn ) is minitive and {vi }ni=1 ∪ {(v1 , . . . , vn )} agree. Statements (i), (ii) ensure that a convex combination of necessities v1 + (1 − )v2 with ∈ (0, 1) is a necessity if and only if v1 , v2 agree. Statement (iii) is of fundamental use since it provides a general way to construct any minitive set function, this matter is exposed in Section 3. Statement (iv) ensures that a necessity can only be a convex combination of minitive set functions. Statement (v) provides a simple way to construct minitive set functions starting from minitive set functions that agree. For instance if v is minitive and : R+ −→ R+ a non-decreasing function with (0) = 0 then (v) is minitive and v, (v) agree. Statements (i), (ii) are particular cases of (v) where is respectively a positive linear transformation and the sum.
3. The general case In order to obtain a decomposition for necessities in the general case we now focus on a simple property of necessities that links them tightly with filter games. One can construct filter games in a natural way when we start from necessities. Let ∈ [0, 1) and v a necessity on (, A), define the -filter Fv = {A ∈ A : v(A) > }, it turns out that Fv is a filter. Similarly if v is a minitive set function = 0, then for all ∈ [0, v()), Fv is a filter. In fact we can fully describe any necessity through its -filters, Proposition 1. Let v : A → [0, 1] be a necessity. Then ∀A ∈ A, v(A) =
[0,1)
uFv (A) d()
(1)
Conversely, let {F }∈[0,1) be a family of filters with F ⊂ F whenever then the set function v(.) = [0,1) uF (.) d() is a necessity measure.
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Moreover,
∀X ∈ B∞ (, A),
X dv =
[0,1)
313
X duF
d(),
(2)
where stands for the usual Lebesgue measure on [0, 1). Proof. For the first part of the proposition, the right-hand side of Eq. (1) is well defined since the function under the integral is monotone. Equality holds since uFv (A) = 1 for 0 < v(A), 0 otherwise. For the second part of the proposition, the set function v is well defined and is minitive by Property (iii). By translation, it suffices to consider the case where X ∈ B∞ (, A), X 0. We have, ∞ ∞ X dv = v({X t}) d(t) = uF ({X t}) d() d(t) 0 0 [0,1) ∞ = uF ({X t}) d(t) d() 0 [0,1) X duF d(). = [0,1)
These equalities hold by Fubini’s theorem the function h : (, t) ∈ [0, 1) × [0, ∞[→ h(, t) = since ∞ v uF ({X t}) is non-negative and that [0,1) 0 h(, t) d(t) d() is finite. We shall refer from now on to (1) as the canonical description of v (see also [5, p. 9], for other representations). We can further specify the canonical description for -continuous necessities, since -filters provide a simple way to check the -continuity of a necessity. Proposition 2. Let v : A → [0, 1] be a necessity. Then v is -continuous if and only if ∀ ∈ [0, 1), Fv is a -filter. Let {F }∈[0,1) be a family of -filters with F ⊂ F whenever then the set function v(.) = [0,1) uF (.) d() is a -continuous necessity. Proof. The first part is obvious from the statement in the Appendix. For the second part. The set function v is well defined and a necessity by Proposition 1. Let A ∈ A and {An }n ⊂ A a non-decreasing sequence converging to . Since for all ∈ [0, 1), uF is -continuous, we have that uF(.) (An ) converges monotoneously to uF(.) () = 1, and by the monotone convergence theorem we get lim v(An ) = lim uF (An ) d() = uF () d() = 1 = v(). ∞
∞
[0,1)
[0,1)
The canonical description of any necessity can be useful since it provides a simple way to approximate a necessity by a (finite) convex combination of its -filters games. Proposition 3. Let v : A → [0, 1] be a necessity. For all N ∈ N, we have: ∀A ∈ A,
N −1 1 1 v (A) − v(A) < . 0 uFk/N N N n=0
(3)
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Moreover, ∀X ∈ B∞ (, A),
0 X 1,
N −1 1 1 v 0 − X dv < . X duFk/N N N
(4)
n=0
v for k = 0, . . . , N − 1. Proof. Let v : A → [0, 1] be a necessity and N ∈ N. Consider the -filters Fk/N N −1 v (A) − v(A) < 1/N . Let be A ∈ A. If v(A) = 0 We have to prove that ∀A ∈ A, 0 1/N n=0 uFk/N or 1 the inequality is verified. Now consider 0 < v(A) < 1. There is kA ∈ [[0, N − 1]] such that v v kA /N < v(A) (kA + 1)/N thus A ∈ Fk/N for k = 0, . . . , kA and A ∈ Fk/N for k = kA + 1, . . . , N − 1 then N −1 1 1 1 v (A) − v(A) = uFk/N (kA + 1) − v(A) < . 0 N N N k=0
Now consider X ∈ B∞ (, A), 0 X 1. Since for all t 0 it holds that 0
N −1 1 1 v ({X t}) − v({X t}) < uFk/N N N k=0
integration on [0, 1] gives N −1 1 1 v X duFk/N 0 − X dv < . N N
k=0
A natural question now arises, can one obtain a Yosida–Hewitt decomposition for a necessity through its canonical description? A natural set to be considered is C = { ∈ [0, 1) : uFv is -continuous}. And if C turns out to be -measurable a possible decomposition would consist of vc = C uFv (.) d() and vp = C c uFv (.) d(). The answer turns out to be affirmative and we can state now a Yosida–Hewitt decomposition theorem for the case of minitive set functions. Theorem 1. Let v be a minitive set function on (, A) then there exists a unique pair of minitive set functions (vc , vp ) with vc -continuous and vp pure such that v = vc + vp . Moreover {v, vc , vp } agree. Proof. From Property (i) it suffices to establish the result for necessities. Let v be a necessity on (, A). Consider the set C = { ∈ [0, 1) : uFv is -continuous}. Existence. If C c = ∅ we are through. So assume C c = ∅. Let ∈ C c , then there exists a non-decreasing sequence {An }n with ∪n An = such that uFv (An ) = 0 < uFv () = 1 i.e. ∀n, An ∈ Fv .
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For , since Fv ⊂ Fv we have ∀n, An ∈ Fv , hence if ∈ C c then [, 1) ⊂ C c . So C c is an interval and C also, hence C c and C are -measurable. Let 0 = Inf C c . We can put now vc = [0,0 ) uFv (.) d() and vp = [0 ,1) uFv (.) d(). vc and vp are minitive set functions comes from Property (iii). That vc is -continuous follows from Proposition 2. In order to prove that vp is pure first we prove that vc is maximal i.e. if v w 0 and w is -continuous then w vc ⇒ w = vc . Take k0 ∈ N such that 0 + 1/k0 < 1, for k k0 , 0 + 1/k ∈ C c , hence there exists a non-decreasing sequence {Akn }n with ∪n Akn = such that ∀ 0 + 1/k, uFv (Akn ) = 0 thus vp (Akn ) 1/k. Now consider a -continuous set function w with v w vc . Let A ∈ A. We have for all n ∈ N, vc (Akn ∩ A) w(Akn ∩ A) v(Akn ∩ A) vc (Akn ∩ A) + 1/k. Since w and vc are -continuous, letting n go to infinity gives vc (A) w(A) vc (A) + 1/k, and since this holds for all k k0 this entails w(A) = vc (A). Let us prove now that vp is pure. Let be a -continuous set function such that 0 vp , we have v = vc + vp vc + vc and vc + is -continuous but now vc is maximal hence vc + = vc thus = 0, so vp is pure. Unicity. Let w1 , w2 be a decomposition of v where w1 is -continuous and w2 is pure and wi are minitive set functions. We will show that w1 = vc and w2 = vp . Since v Max {vc , w1 } vc and Max {vc , w1 } is -continuous, maximality of vc entails vc Max {vc , w1 } that is vc w1 . Now w2 = vc −w1 +vp vc −w1 0 and vc −w1 is -continuous, by purity of w2 we obtain vc −w1 = 0, and w2 = vp follows. Let us check that {v, vc , vp } agree. Since vc and vp and v = vc + vp are minitive set functions, Property (ii) entails that vc and vp agree. Now consider the monotone function : (x, y) → x + y, Property (v) entails that {v, vc , vp } agree. 4. The countable case In this section we shall consider the case where is countable and A is the power set, for sake of simplicity and without loss of generality we can identify and A, respectively, with N and P (N). In this setting a sharper description of the continuous part can be achieved, and a tractable way to compute the Choquet integral is obtained. This provides a natural extension to the infinite-countable case of the characterization of consonant beliefs through a nested family of finite support functions (see [10]). Let Pf (N) denote the set of finite subsets of N.
Theorem 2. Let v be a set function on P (N). If v is -continuous and minitive then there exists a unique (finite or infinite) sequence of subsets {(mn , An )}n with mn > 0 and n mn < +∞ and {An }n ⊂ Pf (N) \ {∅}, ∀n ∈ N, An ⊂ An+1 such that mn uAn . (5) v= n∈N
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Conversely, let {mn }n with mn > 0 and n mn < +∞ and {An }n ⊂ Pf (N) \ {∅} with An ⊂ An+1 then the set function defined by (5) is -continuous and minitive. Moreover, ∀x ∈ B∞ (N, P (N)),
N
x dv =
mn MinAn x.
(6)
n∈N
Proof. Let ∈ [0, v()). Since v is -continuous, Fv is a -filter. Now consider the countable partition {{n}}n∈N . From (iv ) there exists a minimal set A ∈ Pf (N), = ∅ such that A ∈ Fv , hence FA ⊂ Fv . Conversely, if B ∈ Fv then B ∩ A ∈ Fv , but A is minimal thus A ⊂ B ∩ I0 that is A ⊂ B hence B ∈ FA , so Fv ⊂ FA . We have found a set A ∈ Pf (N) such that Fv = FA . Now since Pf (N) is countable there is a countable family of finite subsets An such that {An }n = {A }∈[0,v()) . Moreover since the A are nested i.e. ∀ 0 < < v(), A ⊂ A , there are some mn > 0 with n mn = v() such that ∀ ∈ [0, v()),
Fv
=
FA0 if 0 < m0 , n FAn if n−1 k=0 mk , k=0 mk <
and the canonical description of v through (1) and (2) give (5) and (6). For the converse. That each An is finite ensures the -continuity of each uAn , and since mn 0 and n mn < +∞ the set function v given by (5) is well defined, monotone and -continuous. It remains to prove minitivity of v. Let A, B ⊂ N. Assume v(A), v(B) > v(A ∩ B), then ∃Ap , Aq ∈ {An }n with Ap ⊂ A, Aq ⊂ B and A ∩ B =⊂ Ap , Aq with mp , mq > 0. Since {An }n is a chain, then either Ap ⊂ Aq or Aq ⊂ Ap . Assume Ap ⊂ Aq , then Ap ⊂ A, B but A ∩ B =⊂ Ap , a contradiction. Let x ∈ B∞ (N, P (N)), x 0, then N
x dv =
+∞
+∞
v({x t}) dt =
0
0
=
n∈N
=
mn uAn ({x t}) dt n∈N +∞
mn
0
uAn ({x t}) dt
mn MinAn x
n∈N
which holds by interversion since mn > 0 and v({x .}) = translation the result holds also for x ∈ B∞ (N, P (N)).
n∈N mn uAn ({x .})
is integrable. By
We end this section providing two natural ways to construct necessities which are neither -continuous nor pure. The first one is related to unanimity games whereas the second is achieved via the filter of cofinite sets: co(N) = {A : A ⊂ N, Ac is finite}.
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Example 1. Let K, L be a non-empty set in N where K ⊂ L and K is finite, L infinite. Consider the normalized set function defined as follows: ⎧ ⎨ 0 if K ⊂ A, ∀A ∈ P (N), v(A) = 21 if K ⊂ A and L ⊂ A, ⎩ 1 otherwise. This set function can be written as v = 21 uK + 21 uL . So v is the mean of two necessities that agree, since FL ⊂ FK . Properties (i), (ii) entail that v is a necessity. Moreover since K is finite, uFK is -continuous. And since for all n ∈ N, uL ({0, . . . , n}) = 0, uL is not -continuous. So by Lemma 1 uL must be pure. Therefore, vc = 21 uK and vp = 21 uL . Example 2. Let K be a non-empty finite set in N. Consider the normalized set function defined as follows: ⎧ ⎨ 0 if K ⊂ A, ∀A ∈ P (N), v(A) = 21 if K ⊂ A and Ac is infinite, ⎩ 1 otherwise. This set function can be written as v = 21 uFK + 21 uFK ∩co(N) . So v is the mean of two necessities that agree, since the intersection of two filters is a filter and that FK ∩ co(N) ⊂ FK . Properties (i), (ii) entail that v is a necessity. Moreover since K is finite, uFK is -continuous. And since for all n ∈ N, uFK ∩co(N) ({0, . . . , n}) = 0, uFK ∩co(N) is not -continuous. So by Lemma 1 uFK ∩co(N) must be pure. Therefore, vc = 21 uFK and vp = 21 uFK ∩co(N) . Acknowledgements I wish to thank Professor A. Chateauneuf and D. Denneberg for their helpful advices. The author is also grateful for valuable comments and suggestions for improvements from two anonymous referees. Appendix For sake of completeness we provide a proof of statement in Section 2: Statement. Let v be a monotone convex set function then v is -continuous if and only if v is continuous at . Proof. (only if) is immediate. (if) It suffices to check continuity from above and below. First we start by proving continuity from below. Consider a non-decreasing sequence {An }n ⊂ A converging to A. Putting Bn = An ∪ Ac for all n ∈ N, convexity entails v() + v(An ) v(Bn ) + v(A). Since {Bn }n ⊂ A converges to , continuity at gives lim∞ v(An ) v(A). Equality follows since v is monotone. In order to prove continuity from above consider a non-increasing sequence {An }n ⊂ A converging to A. Putting Bn = A ∪ Acn for all n ∈ N, convexity entails v() + v(A) v(An ) + v(Bn ).
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Since {Bn }n ⊂ A converges to , continuity at gives v(A) lim∞ v(An ). Equality follows since v is monotone. References [1] C. Aliprantis, K. Border, Infinite dimensional analysis: a Hitchhiker’s Guide, second ed., Springer, Berlin, 1999. [2] M. Brüning, D. Denneberg, The sigma-additive Möbius transform of belief measures via Choquet’s theorem, Working paper, Universität Bremen 2003. [3] A. Chateauneuf, Y. Rébillé, A Yosida–Hewitt decomposition for totally monotone games, Math. Social Sci. 48 (2004) 1–9. [4] G. Choquet, Théorie des capacités, in: Annales de l’Institut Fourier `5, Grenoble, 1953–1954, pp. 131–295. [5] D. Denneberg, Non-additive measure and integral, basic concepts and their rôle for applications, in: M. Grabisch, T. Murofushi, M. Sugeno (Eds.), Studies in Fuzzyness and Soft Computing, Fuzzy Measures and Integrals—Theory and Applications, vol. 40, Physica Verlag, Heidelberg, 2000, pp. 42–69. [6] D. Dubois, H. Prade, Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York, 1988. [7] M. Marinacchi, Decomposition and representation of coalitional games, Math. Oper. Res. 21 (1996) 1000–1015. [8] H. Nguyen, Some mathematical tools for linguistic probabilities, Fuzzy Sets and Systems 2 (1979) 53–65. [9] M. Puri, D. Ralescu, A possibility measure is not a fuzzy measure, Fuzzy Sets and Systems 7 (1982) 311–313. [10] G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, 1976. [11] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Institute of Technology, 1974. [12] K. Yosida, E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952) 46–66. [13] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3–28.
A Yosida–Hewitt decomposition for minitive set functions Yann Rébillé∗ Université Paris I, CERMSEM, 106-112 boulevard de l’Hopital, 75647 Paris Cedex 13, France Received 25 August 2004; received in revised form 9 June 2005; accepted 17 June 2005 Available online 11 July 2005
Abstract We prove for minitive set functions defined on a -algebra, a similar decomposition theorem to the Yosida–Hewitt’s one for classical measures, this way any minitive set function can be decomposed in a fuzzy minitive measure part and a purely minitive part. A particular attention is given for the countable case where the canonical description of any -continuous necessity is fully elicited and provide a simple way to compute the Choquet integral of any bounded sequence. © 2005 Elsevier B.V. All rights reserved. Keywords: Non-additive measures; Yosida–Hewitt decomposition; Necessity measures; Choquet integral
1. Introduction A possibility measure [13] is a set function : P () −→ [0, 1] such that (∅) = 0, () = 1 and for all family of subsets {Ai }I it holds, (∪I Ai ) = supI (Ai ), where the index set I is arbitrary. This last property expresses the complete maxitivity of the set function . By duality one can define a necessity measure associated to through N(A) = 1 − (Ac ) (where (.)c denotes the complement). In this case the complete maxitivity translates into complete minitivity i.e. for all family of subsets {Ai }I it holds, N(∩I Ai ) = inf I N(Ai ). A fuzzy measure [11] v : A −→ [0, 1] is a set function defined on a -algebra which satisfies the following conditions v(∅) = 0 , A ⊂ B ⇒ v(A) v(B), A1 ⊂ A2 ⊂ . . . , ⇒ v(∪n An ) = lim∞ v(An ) and A1 ⊃ A2 ⊃ . . . , ⇒ v(∩n An ) = lim∞ v(An ). Puri and Ralescu [9] proved that not all possibilities are fuzzy measures and that possibilities which are fuzzy measures are ∗ Tel.: +33 625454442.
E-mail address: [email protected]. 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.06.012
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ill-behaved on metric spaces. Nguyen [8] suggested for the case of topological space to restrict continuity from above to countable sequences of compact set, this way a Choquet [4] capacity would obtain if we start from an upper semi-continuous density. A natural restriction to the possibility measure is to maintain the maxitivity property only for finite index sets. This way the density function defined for all ∈ through () = ({}) does not summarize all the information. Now in order to obtain continuity from below it is sufficient and necessary to extend the maxitivity property to countable index set, but this last condition does not guarantee continuity from above, thus might not be a fuzzy measure. In the context of classical measure theory [12] provide an important theorem that allows one to decompose any finitely additive set function P into a -additive part and a pathological part termed purely finitely additive i.e. for all -additive set function, 0 P ⇒ = 0. Theorem (Yosida and Hewitt [12]). Let be a finitely additive set function on (, A) then there exists a unique pair of additive set functions (c , p ) with c -additive and p purely finitely additive such that = c + p . Our aim is to give a similar decomposition for the case of necessity measures and decompose any necessity set function into a fuzzy measure part and a purely minitive part. The paper is structured as follows. In Section 2 we give the definition and notations relative to set functions and list some properties of minitive set function that will be used in the sequel. In Section 3 the general case is treated through the use of the canonical description of minitive set functions. In Section 4 we focus on the countable case where the canonical description of any -continuous necessity is fully elicited and provide a simple way to compute the Choquet integral of any bounded sequence.
2. Definitions, notations Let be a non-empty set, and A a -algebra of subsets of . A set function v on A will always be real valued and zero at the empty set i.e. v : A −→ R, v(∅) = 0. Such set functions are known in the field of cooperative games as games. A set function is said to be monotone if ∀A, B ∈ A, A ⊂ B ⇒ v(A) v(B); normalized if v() = 1. A set function v is said to be convex if v is a monotone set function and supermodular i.e. ∀A, B ∈ A, v(A∪B)+v(A∩B) v(A)+v(B). A non-negative set function v (i.e. v 0) is said to be a (finitely) additive set function if ∀A, B ∈ A, A ∩ B = ∅, v(A ∪ B) = v(A) + v(B). If furthermore v() = 1, v is a probability. A set function is said to be -additive if ∀ {An }n ⊂ A, with An ∩ Am = ∅, when m = n, v(∪n An ) = n v(An ). In particular an additive set function is convex. A set function is said to be minitive if ∀A, B ∈ A, v(A ∩ B) = min{v(A), v(B)}. If v is normalized then v is called a necessity (see [6]). It turns out immediately that v is convex. A set function is said to be continuous from below if for all non-decreasing sequences {An }n ⊂ A, lim∞ v(An ) = v(∪n An ). A set function is said to be continuous from above if for all non-increasing sequences {Bn }n ⊂ A, lim∞ v(Bn ) = v(∩n Bn ). v is said to be -continuous if for all sequence {An }n ⊂ A converging to A ∈ A i.e. such that A = ∩n ∪k n Ak = ∪n ∩k n Ak we have lim∞ v(An ) = v(A). If v is a monotone set function then -continuity is equivalent to continuity from below and above. Monotone -continuous set functions are known as fuzzy measures (see [9]). For the case of additive measures these definitions of continuity collapse to -additivity.
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A simple way to obtain -continuity of a set function is through convexity, if v is convex then v is -continuous as soon as v is continuous at i.e. for all non-decreasing sequences {An }n ⊂ A such that ∪n An = , lim∞ v(An ) = v() (see Appendix). The definition of purity for additive set functions can be adapted in the same way to minitive set functions, Definition 2.1. A minitive set function v is said to be pure if for all non-negative -continuous set function, 0 v ⇒ = 0. Let v be a monotone set function on A, define for any X ∈ B∞ (, A) the Choquet integral [4] of X with respect to v as, ∞ 0 X dv = v({X t}) d(t) + (v({X t}) − v()) dt, 0
−∞
where {X t} = { ∈ : X() t} and B∞ (, A) denotes the set of bounded (w.r.t. the supremum norm) A-measurable functions. A family F of subsets of with F ⊂ A is said to be a filter (e.g. [1]) if, (i) ∅ ∈ / F, ∈ F, (ii) ∀A, B ∈ A, [A, B ∈ F ⇒ A ∩ B ∈ F ], (iii) ∀A, B ∈ A, [A ∈ F , A ⊂ B ⇒ B ∈ F ]. A filter is termed -filter if it satisfies the following condition: (iv) for all non-decreasing sequence {An }n with ∪n An = , there exists N ∈ N such that AN ∈ F . Another way to state (iv) is the following, (iv ) Let {An }n ⊂ A be a countable partition i.e. ∀n = m, An ∩ Am = ∅ and ∪n An = then there exists a finite set I ⊂ N such that ∪n∈I An ∈ F . This way if F is a -filter and {An }n ⊂ A a countable partition then one can find a (minimal) finite set I0 such that ∪n∈I0 An ∈ F and ∀I ⊂ N, ∪n∈I An ∈ F ⇒ I0 ⊂ I . Let F be a filter on , define the filter game uF , 1 if B ∈ F , uF (B) = 0 otherwise. For A ∈ A, A = ∅ let FA = {B : B ∈ A, A ⊂ B} we define the unanimity game uA on A by uA := uFA . Notice that for a filter game uF , uF is continuous from below (hence -continuous) if and only if F is a -filter. The nature of a filter game is clear-cut with respect to continuity, Lemma 1. Let uF be a filter game, then uF is pure if and only if uF is not -continuous. Proof. (only if) Let uF be pure and assume uF -continuous, we have uF uF hence uF = 0, what is impossible. (if) Let be -continuous with 0 uF . Since uF is not -continuous, there exists a non-decreasing sequence {An }n with ∪n An = such that ∀n ∈ N, uF (An ) = 0. Now let A ∈ A, for all n ∈ N we have 0 (A ∩ An ) uF (A ∩ An ) uF (An ) = 0 and since {A ∩ An }n converges to A -continuity of entails (A) = 0, so uF is pure.
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Filter games play a keyrole in the integral representation of belief functions (see [7]) since they constitute the set of extreme points of the (convex) set of belief functions (see [2,3]). It appears that they will also play a keyrole in the representation of necessities, since any filter game is a necessity, despite the fact that the set of necessities is not convex. We state some properties that will be useful in the sequel, Properties. The following statements hold, (i) Let v be a minitive set function and > 0 then . v is a minitive set function. (ii) Let v1 , v2 be minitive set functions then v1 + v2 is a minitive set function if and only if v1 , v2 agree i.e. ∃A, B ∈ A s.t. v1 (A) − v1 (B) > 0 and v2 (A) − v2 (B) < 0. (iii) Let I be an interval of [0, 1), and let {v }∈I be minitive set functions such that v v and v , v agree whenever < and supI v () < +∞ then I v d is a minitive set function. (iv) Let v be a minitive set function and v1 , v2 monotone set functions such that v = v1 + v2 then v1 , v2 are minitive set functions. (v) Let {vi }ni=1 be minitive set functions on (, A) where n ∈ N, = 0 that agree i.e. ∀i, j ∈ {1, . . . , n}, vi , vj agree, and a function : (R+ )n −→ R+ with (0, . . . , 0) = 0 and non-decreasing w.r.t. each of his argument then (v1 , . . . , vn ) is a minitive set function. Moreover {vi }ni=1 ∪ {(v1 , . . . , vn )} agree. Proof of Properties. (i) is immediate; (ii) Let v1 , v2 be minitive set functions that agree. Let A, B be in A, we have (v1 (A) − v1 (B))(v2 (A) − v2 (B)) 0 since v1 , v2 agree and minitivity of v1 , v2 gives v1 (A ∩ B) + v2 (A ∩ B) = Min{v1 (A), v1 (B)} + Min{v2 (A), v2 (B)} = Min{v1 (A) + v2 (A), v1 (B) + v2 (B)}. For the converse let A, B be in A. We start from: Min{v1 (A), v1 (B)} + Min{v2 (A), v2 (B)} = Min{v1 (A) + v2 (A), v1 (B) + v2 (B)}. Suppose first that v1 (A) + v2 (A) < v1 (B) + v2 (B). Now if v1 (A) < v1 (B) then v1 (A)+Min{v2 (A), v2 (B)} = v1 (A)+v2 (A) thus v2 (A) v2 (B); if v2 (A) < v2 (B) then Min{v1 (A), v1 (B)} + v2 (A) = v1 (A) + v2 (A) thus v1 (A) v1 (B). Now if v1 (A) + v2 (A) > v1 (B) + v2 (B) we get same conclusion by symmetry. Finally suppose v1 (A) + v2 (A) = v1 (B) + v2 (B), then Min{v1 (A), v1 (B)} + Min{v2 (A), v2 (B)} = v1 (A) + v2 (A) = v1 (B) + v2 (B) thus v1 (A) = v1 (B) and v2 (A) = v2 (B). So in any case we have shown that (v1 (A) − v1 (B))(v2 (A) − v2 (B)) 0. (iii) Let I be an interval of (0, 1), and let {v }∈I be minitive set functions such that v v and v , v agree whenever < and supI v () < +∞. Without loss of generality we fix I = (0, 1) and supI v () = 1. Set v = I v d. Define v0 (A) = 1 and v1 (A) = 0 for all A ∈ A. Now for all A ∈ A we have that ( → v (A)) is a non-increasing function bounded by 0 and 1, thus we can deal with its Riemann integral on [0, 1]. Define for all n ∈ N, = 0 the set function wn by wn = 1/n ni=1 vi/n . These set functions are minitive from (i), (ii) and converge setwise to v. Let be A, B ∈ A. We have, v(A ∩ B) = lim wn (A ∩ B) n
= lim Min{wn (A), wn (B)} n
= Min{lim wn (A), lim wn (B)} n
n
= Min{v(A), V (B)}.
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Now since this equality is obtained for all A, B ∈ A, v is minitive. (iv) Let v be a minitive set function on (, A) and (v1 , v2 ) a decomposition where v1 , v2 are monotone set functions. Let A, B ∈ A, we have v(A ∩ B) = Min{v(A), v(B)} = Min{v1 (A) + v2 (A), v1 (B) + v2 (B)} Min{v1 (A)), v1 (B)} + Min{v2 (A), v2 (B)} v1 (A ∩ B) + v2 (A ∩ B) = v(A ∩ B) thus Min{v1 (A)), v1 (B)} = v1 (A ∩ B) and Min{v2 (A)), v2 (B)} = v2 (A ∩ B) hold. (v) Let {vi }ni=1 be minitive set functions on (, A) where n ∈ N, = 0 that agree, and a function : (R+ )n −→ R+ with (0, . . . , 0) = 0 and non-decreasing w.r.t. each of his argument. Let us show that (v1 , . . . , vn ) is a minitive set function. Let A, B be in A. If for all i ∈ {1, . . . , n} vi (A) = vi (B) holds we are done since vi (A ∩ B) = vi (A) = vi (B). Now assume without loss of generality that v1 (A) < v1 (B). Since {vi }ni=1 agree then for i = 2, . . . , n we have vi (A) vi (B) so by non-decreasingness of it gives (v1 (A), . . . , vn (A)) (v1 (B), . . . , vn (B)) thus (v1 (A∩B), . . . , vn (A∩B)) = (v1 (A), . . . , vn (A)) so (v1 , . . . , vn ) is minitive and {vi }ni=1 ∪ {(v1 , . . . , vn )} agree. Statements (i), (ii) ensure that a convex combination of necessities v1 + (1 − )v2 with ∈ (0, 1) is a necessity if and only if v1 , v2 agree. Statement (iii) is of fundamental use since it provides a general way to construct any minitive set function, this matter is exposed in Section 3. Statement (iv) ensures that a necessity can only be a convex combination of minitive set functions. Statement (v) provides a simple way to construct minitive set functions starting from minitive set functions that agree. For instance if v is minitive and : R+ −→ R+ a non-decreasing function with (0) = 0 then (v) is minitive and v, (v) agree. Statements (i), (ii) are particular cases of (v) where is respectively a positive linear transformation and the sum.
3. The general case In order to obtain a decomposition for necessities in the general case we now focus on a simple property of necessities that links them tightly with filter games. One can construct filter games in a natural way when we start from necessities. Let ∈ [0, 1) and v a necessity on (, A), define the -filter Fv = {A ∈ A : v(A) > }, it turns out that Fv is a filter. Similarly if v is a minitive set function = 0, then for all ∈ [0, v()), Fv is a filter. In fact we can fully describe any necessity through its -filters, Proposition 1. Let v : A → [0, 1] be a necessity. Then ∀A ∈ A, v(A) =
[0,1)
uFv (A) d()
(1)
Conversely, let {F }∈[0,1) be a family of filters with F ⊂ F whenever then the set function v(.) = [0,1) uF (.) d() is a necessity measure.
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Moreover,
∀X ∈ B∞ (, A),
X dv =
[0,1)
313
X duF
d(),
(2)
where stands for the usual Lebesgue measure on [0, 1). Proof. For the first part of the proposition, the right-hand side of Eq. (1) is well defined since the function under the integral is monotone. Equality holds since uFv (A) = 1 for 0 < v(A), 0 otherwise. For the second part of the proposition, the set function v is well defined and is minitive by Property (iii). By translation, it suffices to consider the case where X ∈ B∞ (, A), X 0. We have, ∞ ∞ X dv = v({X t}) d(t) = uF ({X t}) d() d(t) 0 0 [0,1) ∞ = uF ({X t}) d(t) d() 0 [0,1) X duF d(). = [0,1)
These equalities hold by Fubini’s theorem the function h : (, t) ∈ [0, 1) × [0, ∞[→ h(, t) = since ∞ v uF ({X t}) is non-negative and that [0,1) 0 h(, t) d(t) d() is finite. We shall refer from now on to (1) as the canonical description of v (see also [5, p. 9], for other representations). We can further specify the canonical description for -continuous necessities, since -filters provide a simple way to check the -continuity of a necessity. Proposition 2. Let v : A → [0, 1] be a necessity. Then v is -continuous if and only if ∀ ∈ [0, 1), Fv is a -filter. Let {F }∈[0,1) be a family of -filters with F ⊂ F whenever then the set function v(.) = [0,1) uF (.) d() is a -continuous necessity. Proof. The first part is obvious from the statement in the Appendix. For the second part. The set function v is well defined and a necessity by Proposition 1. Let A ∈ A and {An }n ⊂ A a non-decreasing sequence converging to . Since for all ∈ [0, 1), uF is -continuous, we have that uF(.) (An ) converges monotoneously to uF(.) () = 1, and by the monotone convergence theorem we get lim v(An ) = lim uF (An ) d() = uF () d() = 1 = v(). ∞
∞
[0,1)
[0,1)
The canonical description of any necessity can be useful since it provides a simple way to approximate a necessity by a (finite) convex combination of its -filters games. Proposition 3. Let v : A → [0, 1] be a necessity. For all N ∈ N, we have: ∀A ∈ A,
N −1 1 1 v (A) − v(A) < . 0 uFk/N N N n=0
(3)
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Moreover, ∀X ∈ B∞ (, A),
0 X 1,
N −1 1 1 v 0 − X dv < . X duFk/N N N
(4)
n=0
v for k = 0, . . . , N − 1. Proof. Let v : A → [0, 1] be a necessity and N ∈ N. Consider the -filters Fk/N N −1 v (A) − v(A) < 1/N . Let be A ∈ A. If v(A) = 0 We have to prove that ∀A ∈ A, 0 1/N n=0 uFk/N or 1 the inequality is verified. Now consider 0 < v(A) < 1. There is kA ∈ [[0, N − 1]] such that v v kA /N < v(A) (kA + 1)/N thus A ∈ Fk/N for k = 0, . . . , kA and A ∈ Fk/N for k = kA + 1, . . . , N − 1 then N −1 1 1 1 v (A) − v(A) = uFk/N (kA + 1) − v(A) < . 0 N N N k=0
Now consider X ∈ B∞ (, A), 0 X 1. Since for all t 0 it holds that 0
N −1 1 1 v ({X t}) − v({X t}) < uFk/N N N k=0
integration on [0, 1] gives N −1 1 1 v X duFk/N 0 − X dv < . N N
k=0
A natural question now arises, can one obtain a Yosida–Hewitt decomposition for a necessity through its canonical description? A natural set to be considered is C = { ∈ [0, 1) : uFv is -continuous}. And if C turns out to be -measurable a possible decomposition would consist of vc = C uFv (.) d() and vp = C c uFv (.) d(). The answer turns out to be affirmative and we can state now a Yosida–Hewitt decomposition theorem for the case of minitive set functions. Theorem 1. Let v be a minitive set function on (, A) then there exists a unique pair of minitive set functions (vc , vp ) with vc -continuous and vp pure such that v = vc + vp . Moreover {v, vc , vp } agree. Proof. From Property (i) it suffices to establish the result for necessities. Let v be a necessity on (, A). Consider the set C = { ∈ [0, 1) : uFv is -continuous}. Existence. If C c = ∅ we are through. So assume C c = ∅. Let ∈ C c , then there exists a non-decreasing sequence {An }n with ∪n An = such that uFv (An ) = 0 < uFv () = 1 i.e. ∀n, An ∈ Fv .
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For , since Fv ⊂ Fv we have ∀n, An ∈ Fv , hence if ∈ C c then [, 1) ⊂ C c . So C c is an interval and C also, hence C c and C are -measurable. Let 0 = Inf C c . We can put now vc = [0,0 ) uFv (.) d() and vp = [0 ,1) uFv (.) d(). vc and vp are minitive set functions comes from Property (iii). That vc is -continuous follows from Proposition 2. In order to prove that vp is pure first we prove that vc is maximal i.e. if v w 0 and w is -continuous then w vc ⇒ w = vc . Take k0 ∈ N such that 0 + 1/k0 < 1, for k k0 , 0 + 1/k ∈ C c , hence there exists a non-decreasing sequence {Akn }n with ∪n Akn = such that ∀ 0 + 1/k, uFv (Akn ) = 0 thus vp (Akn ) 1/k. Now consider a -continuous set function w with v w vc . Let A ∈ A. We have for all n ∈ N, vc (Akn ∩ A) w(Akn ∩ A) v(Akn ∩ A) vc (Akn ∩ A) + 1/k. Since w and vc are -continuous, letting n go to infinity gives vc (A) w(A) vc (A) + 1/k, and since this holds for all k k0 this entails w(A) = vc (A). Let us prove now that vp is pure. Let be a -continuous set function such that 0 vp , we have v = vc + vp vc + vc and vc + is -continuous but now vc is maximal hence vc + = vc thus = 0, so vp is pure. Unicity. Let w1 , w2 be a decomposition of v where w1 is -continuous and w2 is pure and wi are minitive set functions. We will show that w1 = vc and w2 = vp . Since v Max {vc , w1 } vc and Max {vc , w1 } is -continuous, maximality of vc entails vc Max {vc , w1 } that is vc w1 . Now w2 = vc −w1 +vp vc −w1 0 and vc −w1 is -continuous, by purity of w2 we obtain vc −w1 = 0, and w2 = vp follows. Let us check that {v, vc , vp } agree. Since vc and vp and v = vc + vp are minitive set functions, Property (ii) entails that vc and vp agree. Now consider the monotone function : (x, y) → x + y, Property (v) entails that {v, vc , vp } agree. 4. The countable case In this section we shall consider the case where is countable and A is the power set, for sake of simplicity and without loss of generality we can identify and A, respectively, with N and P (N). In this setting a sharper description of the continuous part can be achieved, and a tractable way to compute the Choquet integral is obtained. This provides a natural extension to the infinite-countable case of the characterization of consonant beliefs through a nested family of finite support functions (see [10]). Let Pf (N) denote the set of finite subsets of N.
Theorem 2. Let v be a set function on P (N). If v is -continuous and minitive then there exists a unique (finite or infinite) sequence of subsets {(mn , An )}n with mn > 0 and n mn < +∞ and {An }n ⊂ Pf (N) \ {∅}, ∀n ∈ N, An ⊂ An+1 such that mn uAn . (5) v= n∈N
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Conversely, let {mn }n with mn > 0 and n mn < +∞ and {An }n ⊂ Pf (N) \ {∅} with An ⊂ An+1 then the set function defined by (5) is -continuous and minitive. Moreover, ∀x ∈ B∞ (N, P (N)),
N
x dv =
mn MinAn x.
(6)
n∈N
Proof. Let ∈ [0, v()). Since v is -continuous, Fv is a -filter. Now consider the countable partition {{n}}n∈N . From (iv ) there exists a minimal set A ∈ Pf (N), = ∅ such that A ∈ Fv , hence FA ⊂ Fv . Conversely, if B ∈ Fv then B ∩ A ∈ Fv , but A is minimal thus A ⊂ B ∩ I0 that is A ⊂ B hence B ∈ FA , so Fv ⊂ FA . We have found a set A ∈ Pf (N) such that Fv = FA . Now since Pf (N) is countable there is a countable family of finite subsets An such that {An }n = {A }∈[0,v()) . Moreover since the A are nested i.e. ∀ 0 < < v(), A ⊂ A , there are some mn > 0 with n mn = v() such that ∀ ∈ [0, v()),
Fv
=
FA0 if 0 < m0 , n FAn if n−1 k=0 mk , k=0 mk <
and the canonical description of v through (1) and (2) give (5) and (6). For the converse. That each An is finite ensures the -continuity of each uAn , and since mn 0 and n mn < +∞ the set function v given by (5) is well defined, monotone and -continuous. It remains to prove minitivity of v. Let A, B ⊂ N. Assume v(A), v(B) > v(A ∩ B), then ∃Ap , Aq ∈ {An }n with Ap ⊂ A, Aq ⊂ B and A ∩ B =⊂ Ap , Aq with mp , mq > 0. Since {An }n is a chain, then either Ap ⊂ Aq or Aq ⊂ Ap . Assume Ap ⊂ Aq , then Ap ⊂ A, B but A ∩ B =⊂ Ap , a contradiction. Let x ∈ B∞ (N, P (N)), x 0, then N
x dv =
+∞
+∞
v({x t}) dt =
0
0
=
n∈N
=
mn uAn ({x t}) dt n∈N +∞
mn
0
uAn ({x t}) dt
mn MinAn x
n∈N
which holds by interversion since mn > 0 and v({x .}) = translation the result holds also for x ∈ B∞ (N, P (N)).
n∈N mn uAn ({x .})
is integrable. By
We end this section providing two natural ways to construct necessities which are neither -continuous nor pure. The first one is related to unanimity games whereas the second is achieved via the filter of cofinite sets: co(N) = {A : A ⊂ N, Ac is finite}.
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Example 1. Let K, L be a non-empty set in N where K ⊂ L and K is finite, L infinite. Consider the normalized set function defined as follows: ⎧ ⎨ 0 if K ⊂ A, ∀A ∈ P (N), v(A) = 21 if K ⊂ A and L ⊂ A, ⎩ 1 otherwise. This set function can be written as v = 21 uK + 21 uL . So v is the mean of two necessities that agree, since FL ⊂ FK . Properties (i), (ii) entail that v is a necessity. Moreover since K is finite, uFK is -continuous. And since for all n ∈ N, uL ({0, . . . , n}) = 0, uL is not -continuous. So by Lemma 1 uL must be pure. Therefore, vc = 21 uK and vp = 21 uL . Example 2. Let K be a non-empty finite set in N. Consider the normalized set function defined as follows: ⎧ ⎨ 0 if K ⊂ A, ∀A ∈ P (N), v(A) = 21 if K ⊂ A and Ac is infinite, ⎩ 1 otherwise. This set function can be written as v = 21 uFK + 21 uFK ∩co(N) . So v is the mean of two necessities that agree, since the intersection of two filters is a filter and that FK ∩ co(N) ⊂ FK . Properties (i), (ii) entail that v is a necessity. Moreover since K is finite, uFK is -continuous. And since for all n ∈ N, uFK ∩co(N) ({0, . . . , n}) = 0, uFK ∩co(N) is not -continuous. So by Lemma 1 uFK ∩co(N) must be pure. Therefore, vc = 21 uFK and vp = 21 uFK ∩co(N) . Acknowledgements I wish to thank Professor A. Chateauneuf and D. Denneberg for their helpful advices. The author is also grateful for valuable comments and suggestions for improvements from two anonymous referees. Appendix For sake of completeness we provide a proof of statement in Section 2: Statement. Let v be a monotone convex set function then v is -continuous if and only if v is continuous at . Proof. (only if) is immediate. (if) It suffices to check continuity from above and below. First we start by proving continuity from below. Consider a non-decreasing sequence {An }n ⊂ A converging to A. Putting Bn = An ∪ Ac for all n ∈ N, convexity entails v() + v(An ) v(Bn ) + v(A). Since {Bn }n ⊂ A converges to , continuity at gives lim∞ v(An ) v(A). Equality follows since v is monotone. In order to prove continuity from above consider a non-increasing sequence {An }n ⊂ A converging to A. Putting Bn = A ∪ Acn for all n ∈ N, convexity entails v() + v(A) v(An ) + v(Bn ).
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Since {Bn }n ⊂ A converges to , continuity at gives v(A) lim∞ v(An ). Equality follows since v is monotone. References [1] C. Aliprantis, K. Border, Infinite dimensional analysis: a Hitchhiker’s Guide, second ed., Springer, Berlin, 1999. [2] M. Brüning, D. Denneberg, The sigma-additive Möbius transform of belief measures via Choquet’s theorem, Working paper, Universität Bremen 2003. [3] A. Chateauneuf, Y. Rébillé, A Yosida–Hewitt decomposition for totally monotone games, Math. Social Sci. 48 (2004) 1–9. [4] G. Choquet, Théorie des capacités, in: Annales de l’Institut Fourier `5, Grenoble, 1953–1954, pp. 131–295. [5] D. Denneberg, Non-additive measure and integral, basic concepts and their rôle for applications, in: M. Grabisch, T. Murofushi, M. Sugeno (Eds.), Studies in Fuzzyness and Soft Computing, Fuzzy Measures and Integrals—Theory and Applications, vol. 40, Physica Verlag, Heidelberg, 2000, pp. 42–69. [6] D. Dubois, H. Prade, Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York, 1988. [7] M. Marinacchi, Decomposition and representation of coalitional games, Math. Oper. Res. 21 (1996) 1000–1015. [8] H. Nguyen, Some mathematical tools for linguistic probabilities, Fuzzy Sets and Systems 2 (1979) 53–65. [9] M. Puri, D. Ralescu, A possibility measure is not a fuzzy measure, Fuzzy Sets and Systems 7 (1982) 311–313. [10] G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, 1976. [11] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Institute of Technology, 1974. [12] K. Yosida, E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952) 46–66. [13] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3–28.